Sparse identification of differential equations aims to compute the analytic expressions from the observed data explicitly. However, there exist two primary challenges. Firstly, it exhibits sensitivity to the noise in the observed data, particularly for the derivatives computations. Secondly, existing literature predominantly concentrates on single-fidelity (SF) data, which imposes limitations on its applicability due to the computational cost. In this paper, we present two novel approaches to address these problems from the view of uncertainty quantification. We construct a surrogate model employing the Gaussian process regression (GPR) to mitigate the effect of noise in the observed data, quantify its uncertainty, and ultimately recover the equations accurately. Subsequently, we exploit the multi-fidelity Gaussian processes (MFGP) to address scenarios involving multi-fidelity (MF), sparse, and noisy observed data. We demonstrate the robustness and effectiveness of our methodologies through several numerical experiments.

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In this work, we present a new numerical framework for the efficient solution of the time-harmonic elastic wave equation at multiple frequencies. We show that multiple frequencies (and multiple right-hand sides) can be incorporated when the discretized problem is written as a matrix equation. This matrix equation can be solved efficiently using the preconditioned IDR(s) method. We present an efficient and robust way to apply a single preconditioner using MSSS matrix computations. For 3D problems, we present a memory-efficient implementation that exploits the solution of a sequence of 2D problems. Realistic examples in two and three spatial dimensions demonstrate the performance of the new algorithm.@en

This paper studies a new preconditioning technique for sparse systems arising from discretized partial differential equations in computational fluid dynamics problems. This preconditioning technique exploits the multilevel sequentially semiseparable (MSSS) structure of the system matrix. MSSS matrix computations give a data-sparse way to approximate the LU factorization of a sparse matrix from discretized partial differential equations in linear computational complexity with respect to the problem size. In contrast to the standard block diagonal and block upper-triangular preconditioners, we exploit the global MSSS structure of the 2×2 block system from the discretized Stokes equation and linearized Navier-Stokes equation. This avoids approximating the Schur complement explicitly, which is a big advantage over standard block preconditioners. Through numerical experiments on standard computational fluid dynamics benchmark problems in Incompressible Flow and Iterative Solver Software, we show the performance of the MSSS preconditioners. They indicate that the global MSSS preconditioner not only yields mesh size independent convergence but also gives viscosity parameter and Reynolds number independent convergence. Compared with the algebraic multigrid (AMG) method and the geometric multigrid (GMG) method for block preconditioners, the MSSS preconditioning technique is more robust than both the AMG method and GMG method, and considerably faster than the AMG method. Copyright © 2015 John Wiley & Sons, Ltd. @en